The field \(\phi \)  here
is a Higgs-like scalar field,
being a relic of
a Yang-Mills (YM) connection in higher dimensions,
and
approaches a nonzero value at infinity,
with two discontinuous vacua at \(\phi _0=\pm 1\) .
As we shall see, the expression (REF )
of the coupling function
allows for scalarization of Schwarzschild BHs;
the basic properties of the
scalarized
solutions are rather similar to those in Ref. [1]}
(where a quadratic coupling function has been employed).
However,
there are also novel features;
an interesting one is that
scalarization occurs for both signs of the constant \(\alpha \) 
(which is not the case for other models with scalarized static BHs).
Another interesting feature is the existence
of an extension of the model with a vector field, which allows for
vectorization of the generic Einstein-GB-scalar (EGBs) BHs.
